DLMF:8.11.E7 (Q2578): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: incomplete gamma function / rank
 
Normal rank
Property / Symbols used: incomplete gamma function / qualifier
 
Defining formula:

Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}

\incGamma@{\NVar{a}}{\NVar{z}}
Property / Symbols used: incomplete gamma function / qualifier
 
xml-id: C8.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11332 / rank
 
Normal rank
Property / Symbols used: Q11332 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11332 / qualifier
 
xml-id: C8.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11323 / rank
 
Normal rank
Property / Symbols used: Q11323 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11323 / qualifier
 
xml-id: C8.S1.XMD2.m1edec
Property / Symbols used
 
Property / Symbols used: Q11324 / rank
 
Normal rank
Property / Symbols used: Q11324 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11324 / qualifier
 
xml-id: C8.S1.XMD3.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11330 / rank
 
Normal rank
Property / Symbols used: Q11330 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11330 / qualifier
 
xml-id: C8.S1.XMD5.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11343 / rank
 
Normal rank
Property / Symbols used: Q11343 / qualifier
 
Defining formula:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11343 / qualifier
 
xml-id: C8.S1.XMD7.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11344 / rank
 
Normal rank
Property / Symbols used: Q11344 / qualifier
 
Defining formula:

λ 𝜆 {\displaystyle{\displaystyle\lambda}}

\lambda
Property / Symbols used: Q11344 / qualifier
 
xml-id: C8.S11.XMD3.m1adec
Property / Symbols used
 
Property / Symbols used: Q11345 / rank
 
Normal rank
Property / Symbols used: Q11345 / qualifier
 
Defining formula:

b k ⁢ ( λ ) subscript 𝑏 𝑘 𝜆 {\displaystyle{\displaystyle b_{k}(\lambda)}}

b_{k}(\lambda)
Property / Symbols used: Q11345 / qualifier
 
xml-id: C8.S11.XMD4.m1adec

Latest revision as of 13:54, 2 January 2020

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English
DLMF:8.11.E7
No description defined

    Statements

    Defining formula
    Γ ⁡ ( a , z ) ∼ z a ⁢ e - z ⁢ ∑ k = 0 ∞ ( - a ) k ⁢ b k ⁢ ( λ ) ( z - a ) 2 ⁢ k + 1 , asymptotic-expansion incomplete-Gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 superscript 𝑎 𝑘 subscript 𝑏 𝑘 𝜆 superscript 𝑧 𝑎 2 𝑘 1 {\displaystyle{\displaystyle\Gamma\left(a,z\right)\sim z^{a}e^{-z}\sum_{k=0}^{% \infty}\frac{(-a)^{k}b_{k}(\lambda)}{(z-a)^{2k+1}},}}
    0 references
    DLMF id
    DLMF:8.11.E7
    0 references
    constraint
    | ph ⁡ a | ≤ 3 ⁢ π 2 - δ phase 𝑎 3 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta}}
    0 references
    λ > 1 𝜆 1 {\displaystyle{\displaystyle\lambda>1}}
    0 references
    | ph ⁡ a | ≤ 3 ⁢ π 2 - δ ph 𝑎 3 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta}}
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    incomplete gamma function
    Defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
    0 references
    Q11332
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S1.XMD1.m1adec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1edec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1fdec
    0 references
    Q11330
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1ddec
    0 references
    Q11343
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C8.S1.XMD7.m1cdec
    0 references
    Q11344
    Defining formula
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C8.S11.XMD3.m1adec
    0 references
    Q11345
    Defining formula
    b k ⁢ ( λ ) subscript 𝑏 𝑘 𝜆 {\displaystyle{\displaystyle b_{k}(\lambda)}}
    xml-id
    C8.S11.XMD4.m1adec
    0 references
     
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